Particle displacement

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Sound measurements
Characteristic
Symbols
 Sound pressure  p, SPL
 Particle velocity  v, SVL
 Particle displacement  ξ
 Sound intensity  I, SIL
 Sound power  P, SWL
 Sound energy  W
 Sound energy density  w
 Sound exposure  E, SEL
 Acoustic impedance  Z
 Speed of sound  c
 Audio frequency  AF
 Transmission loss  TL

Particle displacement or displacement amplitude is a measurement of distance of the movement of a particle from its equilibrium position in a medium as it transmits a sound wave.[1] The SI unit of particle displacement is the metre (m). In most cases this is a longitudinal wave of pressure (such as sound), but it can also be a transverse wave, such as the vibration of a taut string. In the case of a sound wave travelling through air, the particle displacement is evident in the oscillations of air molecules with, and against, the direction in which the sound wave is travelling.[2]

A particle of the medium undergoes displacement according to the particle velocity of the sound wave traveling through the medium, while the sound wave itself moves at the speed of sound, equal to 343 m/s in air at 20 °C.

Mathematical definition[edit]

Particle displacement, denoted ξ, is given by[3]

\mathbf \xi = \int_{t} \mathbf v\, \mathrm{d}t

where v is the particle velocity.

Progressive sine waves[edit]

The particle displacement of a progressive sine wave is given by

\xi(\mathbf{r},\, t) = \xi_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\xi, 0}),

where

It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave x are given by

v(\mathbf{r},\, t) = \frac{\partial \xi}{\partial t} (\mathbf{r},\, t) = \omega \xi_\mathrm{m} \cos\!\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\xi, 0} + \frac{\pi}{2}\right) = v_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{v, 0}),
p(\mathbf{r},\, t) = -\rho c^2 \frac{\partial \xi}{\partial x} (\mathbf{r},\, t) = \rho c^2 k_x \xi_\mathrm{m} \cos\!\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\xi, 0} + \frac{\pi}{2}\right) = p_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{p, 0}),

where

  • vm is the amplitude of the particle velocity;
  • \varphi_{v, 0} is the phase shift of the particle velocity;
  • pm is the amplitude of the acoustic pressure;
  • \varphi_{p, 0} is the phase shift of the acoustic pressure.

Taking the Laplace transforms of v and p with respect to time yields

\hat{v}(\mathbf{r},\, s) = v_\mathrm{m} \frac{s \cos \varphi_{v,0} - \omega \sin \varphi_{v,0}}{s^2 + \omega^2},
\hat{p}(\mathbf{r},\, s) = p_\mathrm{m} \frac{s \cos \varphi_{p,0} - \omega \sin \varphi_{p,0}}{s^2 + \omega^2}.

Since \varphi_{v,0} = \varphi_{p,0}, the amplitude of the specific acoustic impedance is given by

z_\mathrm{m}(\mathbf{r},\, s) = |z(\mathbf{r},\, s)| = \left|\frac{\hat{p}(\mathbf{r},\, s)}{\hat{v}(\mathbf{r},\, s)}\right| = \frac{p_\mathrm{m}}{v_\mathrm{m}} = \frac{\rho c^2 k_x}{\omega}.

Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by

\xi_\mathrm{m} = \frac{v_\mathrm{m}}{\omega},
\xi_\mathrm{m} = \frac{p_\mathrm{m}}{\omega z_\mathrm{m}(\mathbf{r},\, s)}.

See also[edit]

References and notes[edit]

  1. ^ Julian W. Gardner, V. K. Varadan, Osama O. Awadelkarim (2001). Microsensors, MEMS, and Smart Devices. John Wiley and Sons. pp. 321–322. ISBN 978-0-471-86109-6. 
  2. ^ Arthur Schuster (1904). An Introduction to the Theory of Optics. London: Edward Arnold. 
  3. ^ John Eargle (January 2005). The Microphone Book: From mono to stereo to surround – a guide to microphone design and application. Burlington, Ma: Focal Press. p. 27. ISBN 978-0-240-51961-6. 

Related Reading:

External links[edit]